Dirichlet Series and Euler Products

Abstract

In 1737 Euler proved Euclid’s theorem on the existence of infinitely many primes by showing that the series ∑p ^-1, extended over all primes, diverges. He deduced this from the fact that the zeta function ζ(s), given by

$$\zeta (s) = \sum\limits_{n = 1}^\infty {\frac{1}{{{n^s}}}}$$

((1))

for real s > 1, tends to ∞ as s → 1. In 1837 Dirichlet proved his celebrated theorem on primes in arithmetical progressions by studying the series

$$L(s,\chi ) = \sum\limits_{n = 1}^\infty {\tfrac{{\chi (n)}}{{{n^s}}}}$$

((2))

where χ is a Dirichlet character and s > 1.